partial differential equation

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{Solve.}}\)

\(\displaystyle a\frac{\partial w}{\partial x} + b\frac{\partial w}{\partial y} = cw^n + sw^m\)

 

[logistic_guy]

\(\displaystyle \frac{dx}{a} = \frac{dy}{b} = \frac{dw}{cw^n + sw^m}\)


\(\displaystyle b \ dx = a \ dy\)

\(\displaystyle \int b \ dx = \int a \ dy\)

\(\displaystyle bx = ay + C_1\)

\(\displaystyle bx - ay = C_1\)

And we introduce an arbitrary function \(\displaystyle \Phi(C_1)\):

\(\displaystyle \Phi(C_1) = \Phi(bx - ay) = C_2\)

 

[logistic_guy]

What a lovely day!

\(\displaystyle \frac{dx}{a} = \frac{dy}{b} = \frac{dw}{cw^n + sw^m}\)

\(\displaystyle \frac{dx}{a} = \frac{dw}{cw^n + sw^m}\)


\(\displaystyle \int \frac{dx}{a} = \int \frac{dw}{cw^n + sw^m}\)


\(\displaystyle x + C_2 = a\int \frac{dw}{cw^n + sw^m}\)


Then, the general solution to the partial differential equation is:

\(\displaystyle a\int \frac{dw}{cw^n + sw^m} = x +\Phi(bx - ay)\)